An upper bound on the number of
self-avoiding polygons via joining
For and even, let denote the number of length self-avoiding polygons in up to translation. The polygon cardinality grows exponentially, and the growth rate is called the connective constant and denoted by . Madras [J. Statist. Phys. 78 (1995) no. 3–4, 681–699] has shown that in dimension . Here we establish that for a set of even of full density when . We also consider a certain variant of self-avoiding walk and argue that, when , an upper bound of holds on a full density set for the counterpart in this variant model of this normalized polygon cardinality.
Key words and phrases:
Self-avoiding walk was introduced in the 1940s by Flory and Orr [12, 26] as a model of a long polymer chain in a system of such chains at very low concentration. It is well known among the basic models of discrete statistical mechanics for posing problems that are simple to state but difficult to solve. Two recent surveys are the lecture notes  and [19, Section 3].
1.1. The model
We will denote by the set of non-negative integers. Let . For , let denote the Euclidean norm of . A walk of length with is a map such that for each . An injective walk is called self-avoiding. A walk is said to close (and to be closing) if . When the missing edge connecting and is added, a polygon results.
Let be a closing self-avoiding walk. For , let denote the unordered nearest neighbour edge in with endpoints and . Let denote ’s missing edge, with endpoints and . We call the collection of edges the polygon of . A self-avoiding polygon in is defined to be any polygon of a closing self-avoiding walk in . The polygon’s length is its cardinality.
We will usually omit the adjective self-avoiding in referring to walks and polygons. Recursive and algebraic structure has been used to analyse polygons in such domains as strips, as  describes.
Note that the polygon of a closing walk has length that exceeds the walk’s by one. Polygons have even length and closing walks, odd.
1.2. Main results
Define the polygon number to be the number of length polygons up to translation. By (3.2.9) of , exists; it called the connective constant and denoted by . We define the real-valued polygon number deficit exponent according to the formula
That is non-negative in any dimension will be reviewed in Section 3. Madras  has proved a bound on the moment generating function of the sequence which when would assert were this limit known to exist. More relevantly for us, he has shown in  using a polygon joining technique that for . We develop this technique to prove a stronger lower bound valid for typical high .
The limit supremum density of a set of even, or odd, integers is
When the corresponding limit infimum density equals the limit supremum density, we naturally call it the limiting density.
Let . For any , the limiting density of the set of for which is equal to one.
As we will discuss shortly, in the strongest form of this result consistent with predictions, the value would replace .
Madras and Slade [22, Theorem 6.1.3] have proved that in dimension for spread-out models, in which the vertices of are connected by edges below some bounded distance. There is a prospect that the method of proof of Theorem 1.3 may be applied in all dimensions , and the expected conclusion is that for a full density set of even . Indeed, our second main conclusion, Theorem 1.5, is a result to this effect. Certain technical difficulties arise as we try to apply our method in the higher dimensional case and, in order to largely circumvent these, we present Theorem 1.5 for a variant model.
The maximal edge local time of a nearest neighbour walk is the maximum number of times that traverses an edge of ; more formally, it is the maximum cardinality of a subset such that the unordered sets and coincide for each pair . Call -edge self-avoiding if its maximal edge local time is at most . Note that even -edge self-avoiding walk satisfies a weaker avoidance constraint than does self-avoiding walk.
When considering (as we will) -edge self-avoiding walks, we say that a walk as above closes if . Two such walks may be identified if they coincide after reparametrization by cyclic shift or reversal. A -edge self-avoiding polygon is an equivalence class under this relation on closing walks. The length of such a polygon is the length of any of its members (and is if one of these members is as above). Note that these definitions entail that not only polygons but also closing walks have even length.
For , let denote the number of -edge self-avoiding polygons of length up to translation. The connective constant also exists for this model and we denote it by . We define a real sequence so that
Let . For any , the set of for which has limiting density equal to one.
1.3. Corollaries of the main results: the closing probability
Let denote the set of self-avoiding walks of length that start at , i.e., with . We denote by the uniform law on . The walk under the law will be denoted by . The closing probability is .
Let the walk number equal the cardinality of . By equations (1.2.10) and (3.2.9) of , the limit exists and coincides with , and we have .
The closing probability may be written in terms of the polygon and walk numbers. There are closing walks whose polygon is a given polygon of length , since there are choices of missing edge and two of orientation. Thus,
Let . For any , the set of such that
has limiting density equal to one.
The walk (and polygon) cardinalities, and the closing probability, are notions that may be formulated for our variant model. Indeed, we may write and for the cardinality of, and uniform law on, the set of length -edge self-avoiding walks beginning at . As we will explain in Section 5, we also have . The counterpart
Let . For any , the set of for which
has limiting density equal to one.
1.4. Further applications of the main results
Let . For any and sufficiently high,
The snake method is a probabilistic tool that is introduced in  in order to obtain Theorem 1.8. (The article  is in large part a reworking of , although this term for the method was not used in the original article .) The snake method is applied via a technique of Gaussian pattern fluctuation in  to obtain Theorem 1.8. A second application of the snake method is made in  in order to reach a stronger inference regarding the closing probability, valid when , for a subsequence of even . In this case, the snake method is allied not with a pattern fluctuation technique but instead with the tool that is central to the proofs of the present article’s principal results Theorems 1.3 and 1.5: the polygon joining technique, initiated by Madras in . The conclusion that is reached in  is now stated.
For any , the bound
holds on a set of of limit supremum density at least .
(When , (1.4) should be interpreted as asserting a superpolynomial decay in for the left-hand side.)
The above result not only shares an important common element with Theorem 1.3 in the form of the technique used to prove the two results; the proof of Theorem 1.9 also uses the statement of Theorem 1.3.
It may be apparent that this paper shares with  and  certain combinatorial and probabilistic elements. The present article, and the other two, may be read alone, but there may be also be value in viewing the results in unison. The online article  presents the content of the three articles in a single work and includes some expository discussion. The online work connects the ideas; for example, its Section 3.3 elaborates the upcoming heuristic derivation of the polygon number deficit exponent lower bound (3.1) and uses this derivation as a mnemonic to discuss proofs of several of the main conclusions.
1.5. Exponent prediction and hyperscaling relation
The limiting value is predicted to exist and to satisfy a relation with the Flory exponent for mean-squared radius of gyration. The latter exponent is specified by the putative formula
where denotes the expectation associated with (and where note that is the non-origin endpoint of ); in essence, is supposed to be typically of order . The hyperscaling relation that is expected to hold between and is
where the dimension is arbitrary. In , and thus is expected. That was predicted by the Coulomb gas formalism [24, 25] and then by conformal field theory [10, 11]. Hara and Slade [16, 17] used the lace expansion to show that when by demonstrating that, for some constant , is . This value of is anticipated in four dimensions as well, since is expected to grow as . In fact, the continuous-time weakly self-avoiding walk in has been the subject of an extensive recent investigation of Bauerschmidt, Brydges and Slade. In , a correction to the susceptibility is derived, relying on rigorous renormalization group analysis developed in a five-paper series [5, 6, 3, 7, 8].
The structure of the paper. The paper has four further sections.
The first sets up notation. The second, Section 3, gives an expository overview of the polygon joining technique and, we hope, provides a useful background for reading the later proofs; it is not logically needed later except for gathering some standard facts that will be used. Section 4 proves Theorem 1.3. The argument is varied in Section 5 in order to prove Theorem 1.5.
Acknowledgments. I am very grateful to a referee for a thorough discussion of the article . Indeed, the present form of the two main theorems in the present article is possible on the basis of a suggestion made by this referee, (and this strengthened form has led to an improvement in Theorem 1.9(1), proved in ). I thank Hugo Duminil-Copin and Ioan Manolescu for many stimulating and valuable conversations about the central ideas in the paper. I thank Wenpin Tang for useful comments on a draft version. I would also like to thank Itai Benjamini for suggesting the problem of studying upper bounds on the closing probability.
This work was supported by NSF grant DMS-.
2. Some general notation and tools
2.0.1. Multi-valued maps
For a finite set , let denote its power set. Let be another finite set. A multi-valued map from to is a function . An arrow is a pair for which ; such an arrow is said to be outgoing from and incoming to . We consider multi-valued maps in order to find lower bounds on , and for this, we need upper (and lower) bounds on the number of incoming (and outgoing) arrows. This next lemma is an example of such a lower bound.
Let . Set to be the minimum over of the number of arrows outgoing from , and to be the maximum over of the number of arrows incoming to . Then .
Proof. The quantities and are upper and lower bounds on the total number of arrows. ∎
2.0.2. Denoting walk vertices and subpaths
For with , we write for . For a walk and , we write in place of . For , denotes the subpath given by restricting .
2.0.3. Notation for certain corners of polygons
The most important ideas in the article may be communicated by considering the two-dimensional case via the proof of Theorem 1.3. The proof of Theorem 1.5 is a variation. We now present notation specific to the two-dimensional case.
The Cartesian unit vectors are denoted by and and the coordinates of by and . For a finite set of vertices , we define the northeast vertex in to be that element of of maximal -coordinate; should there be several such elements, we take to be the one of maximal -coordinate. That is, is the uppermost element of , and the rightmost among such uppermost elements if there are more than one. Using the four compass directions, we may similarly define eight elements of , including the lexicographically minimal and maximal elements of , and . We extend the notation to any self-avoiding walk or polygon , writing for example for , where is the vertex set of . For a polygon or walk , set , , and . The height of is and its width is .
2.0.4. Polygons with northeast vertex at the origin
For , let denote the set of length polygons such that . The set is in bijection with equivalence classes of length polygons where polygons are identified if one is a translate of the other. Thus, .
We write for the uniform law on . A polygon sampled with law will be denoted by , as a walk with law is.
There are ways of tracing the vertex set of a polygon of length : choices of starting point and two of orientation. We now select one of these ways. Abusing notation, we may write as a map from to , setting , , and successively defining to be the previously unselected vertex for which and form the vertices incident to an edge in , with the final choice being made. Note that .
2.0.5. Cardinality of a finite set
This is denoted by either or .
The shortest non-empty polygons contain four edges. Certain such polygons play an important role in several arguments and we introduce notation for them now.
A plaquette is a polygon with four edges. Let be a polygon. A plaquette is called a join plaquette of if and intersect at precisely the two horizontal edges of . (The boundaries of the three shaded red squares of the polygon in the upcoming Figure 3 are join plaquettes of the polygon.) Note that when is a join plaquette of , the operation of removing the two horizontal edges in from and then adding in the two vertical edges in to results in two disjoint polygons whose lengths sum to the length of . We use symmetric difference notation and denote the output of this operation by .
The operation may also be applied in reverse: for two disjoint polygons and , each of which contains one vertical edge of a plaquette , the outcome of removing ’s vertical edges and adding in its horizontal ones is a polygon whose length is the sum of the lengths of and .
3. Polygon number bounds via joining: an heuristic prelude
In Section 2.1 of the text , a presentation is made of the standard heuristic derivation of the relation (1.6): note that equation (1.4.14) of  is a representation of (1.6) written in terms of the exponent .
To make the derivation, we hypothesise the existence of the limit
Step A: . By [22, Theorem 3.2.3], we have that, for ,
Thus, the sequence , , is subadditive, so that Fekete’s lemma [27, Lemma 1.2.1] implies the existence of and the bound
Thus, , completing step A. This step thus depends principally on the bound (3.3), which is proved by a simple use of polygon joining. Our aim is to overview ideas for the upcoming proofs of our principal results, and we consider only the case of and equal polygon length in explaining (3.3). Consider a pair . Relabel by translating it so that is one unit to the right of . The plaquette whose upper left vertex is has one vertical edge in and one in . Note that is a polygon of length which is either an element of or may be associated to such an element by making a translation. Moreover, the application is injective, because from we can detect the location of the plaquette . The reader may wish to confirm this property, using that and have the same length. This injectivity implies that .
Step B: . This step is an argument of Madras in . When , we will argue that
where recall that (1.5) specifies . That follows directly (provided the two exponents exist). The derivation of (3.5) develops the polygon joining argument in step A. The length polygons and were joined in only one alignment, after displacement of to a given location. Madras argues under (1.5) that there are at least an order of locations to which may be translated and then attached to . A total of distinct length polygons results, and we obtain (3.5).
Where are these new locations for joining? Orient and so that the height of each is at least its width; thus each height is at least of order . Translate vertically so that some vertices in and share their -coordinate, and then push to the right if need be so that this polygon is to the right of . Then push back to the left stopping just before the two polygons overlap. The two polygons contain vertices at distance one, and so it is plausible that in the locale of this vertex pair, we may typically find a plaquette whose left and right vertical edges are occupied by and . The operation in the proof of step A applied with plaquette then yields a length polygon. This construction began with a vertical translation of , with different choices of this translation resulting in different outcomes for the joined polygon; since there is an order of different heights that may be used for this translation, we see that the bound (3.5) results.
There is in fact a technical difficulty in implementing this argument: in some configurations, no such plaquette exists. Madras developed a local joining procedure that overcomes this difficulty in two dimensions. His procedure will be reviewed shortly, in Section 4.1.
To argue for (3.6), note that, in deriving , each polygon pair resulted in distinct length polygons. The length pair may be varied to be of the form for any . We are constructing a multi-valued map
to the power set of which associates to each polygon pair in the domain an order of elements of . Were injective, we would obtain (3.6). (The term injective is being misused: we mean that no two arrows of are incoming to the same element of .) The map is not injective but it is plausible that it only narrowly fails to be so: that is, abusing notation in a similar fashion, for typical , the cardinality of is at most . A definition is convenient before we argue this.
Let be a polygon, and let be one of ’s join plaquettes. Let and denote the disjoint polygons of which is comprised. If each of and has at least one quarter of the length of , then we call a macroscopic join plaquette.
To see that is close to being injective, note that each pre-image of under corresponds to a macroscopic join plaquette of . Each macroscopic join plaquette entails a probabilistically costly macroscopic four-arm event, where four walks of length of order must approach the plaquette without touching each other. That belongs to amounts to saying that is an element of having at least one macroscopic join plaquette. The four-arm costs make it plausible that a typical such polygon has only a few such plaquettes, gathered together in a small neighbourhood. Thus, is plausibly close to injective so that (3.6), and (3.1), results. (We will later be faced with arguing that is indeed close to injective; although we will obtain a version of (3.6) via a form of near-injectivity, we will not obtain this near-injectivity by making rigorous the above, rather vague, heuristic.)
4. Polygon joining when , rigorously: proving Theorem 1.3
In this section, we prove Theorem 1.3, in which recall that . Our job is clear in light of the preceding heuristic derivation: we must make step rigorous. The next proposition is a key step. It offers a rigorous interpretation of (3.6).
For , the set of -high polygon number indices is given by
For any , there is a constant such that, for ,
where is chosen so that .
In essence, the meaning of the proposition is that when , is at least a small constant multiple of , where the sum is over an interval of order indices around the value (though such a statement does not follow directly from the proposition).
Perhaps our proof of Theorem 1.3 via Proposition 4.2 can be refined to quantify the rate of convergence of the density one index set in the theorem. However, the proposition in isolation is inadequate for proving the conclusion for all , because this tool permits occasional spikes in the value of the , as the sequence
This section has five subsections. The first four present tools needed for the proof of Proposition 4.2, with the proof of this result in the fourth. The fifth proves Theorem 1.3 as a consequence of the proposition.
Proving Proposition 4.2 amounts to making step C rigorous. We need to define in precise terms the polygon joining technique that we will use to do this, and, in the first subsection, we specify the local details of the technique due to Madras that we will use. The heuristic argument in favour of (3.6) depended on the near-injectivity of the multi-valued map , which was argued by making a case for the sparsity of macroscopic join plaquettes. In the second subsection, we present Proposition 4.5 and Corollary 4.6, our rigorous versions of this sparsity claim. In the third subsection, we set up the apparatus needed to specify our joining mechanism , and in the fourth, we define and analyse the mechanism (and so obtain Proposition 4.2).
4.1. Madras’ polygon joining procedure
When a pair of polygons is close, there may not be a plaquette whose vertical edges are divided between the two elements of the pair. We now recall Madras’ joining technique which works in a general way for such pairs.
Consider two polygons and of lengths and for which the intervals
Madras’ procedure joins and to form a new polygon of length in the following manner.
First translate to the right by far enough that the -coordinates of the vertices of this translate are all strictly greater than all of those of . Now shift to the left step by step until the first time at which there is a pair of vertices, one in and the other in the -translate, that share an -coordinate and whose -coordinates differ by at most two; such a moment necessarily occurs, by the assumption (4.2). Write (with ) for this particular horizontal translate of . There is at least one vertex such that the set contains a vertex of and a vertex of . The set of such vertices contains at most one vertex with any given -coordinate. Denote by the vertex with the maximal -coordinate.
Madras now defines a modified polygon , which is formed from by changing its structure in a neighbourhood of . Depending on the structure of near , either two edges are removed and ten edges added to form from , or one edge is removed and nine are added. As such, has length . The rule that specifies is recalled from  in Figure 1.
A modified polygon formed from is also defined. Rotate about the vertex by radians to form a new polygon . Form according to the same rules, recalled in Figure 1. Then rotate back the outcome by radians about to produce the modification of , which to simplify notation we denote by .
Writing in place of , note that no vertex of belongs to the right corridor , the region that lies strictly to the right of . (Indeed, it is this fact that implies that is a polygon.) Equally, no vertex of belongs to the left corridor .
Note that the polygon extends to the right of by either two or three units inside the right corridor (by two in case IIa, IIci or IIIci and by three otherwise). Likewise extends to the left of by either two or three units in the left corridor (by two when satisfies case IIa, IIci or IIIci and by three otherwise).
Note from Figure 1 that, in each case, contains two vertical edges that cross the right corridor at the maximal -coordinate adopted by vertices in that lie in this corridor. Likewise, contains two vertical edges that cross the left corridor at the minimal -coordinate adopted by vertices in that lie in the left corridor.
Translate to the right by units, where equals
five when one of cases IIa, IIci and IIIci obtains for both and ;
six when one of these cases obtains for exactly one of these polygons;
seven when none of these cases holds for either polygon.
Note that and are disjoint polygons such that, for some pair of vertically adjacent plaquettes in the right corridor (whose left sides have -coordinate either or ), the edge-set of intersects the plaquette pair on the two left sides of and , while the edge-set of intersects this pair on the two right sides of and . Let denote the upper element of this plaquette pair.
The polygon that Madras specifies as the join of and is given by . Note that, to form the join polygon, is first horizontally translated by units to form , modified locally to form , and then further horizontally translated by units to produce the polygon that is joined onto . Thus, undergoes a horizontal shift by units as well as a local modification before being joined with .
For two polygons and satisfying (4.2), define the Madras join polygon
The plaquette will be called the junction plaquette.
Such polygons and are called Madras joinable if : that is, no horizontal shift is needed so that may be joined to by the above procedure. Note that the modification made is local in this case: contains at most twenty edges. See Figure 2.
4.2. Global join plaquettes are few
Recall that in step C of the derivation of (3.1), the near-injectivity of the multi-valued map was argued as a consequence of the sparsity of macroscopic join plaquettes. We now present in Corollary 4.6 a rigorous counterpart to this sparsity assertion. In the rigorous approach, we use a slightly different definition to the notion of macroscopic join plaquette.
For , let . A join plaquette of is called global if the two polygons comprising may be labelled and in such a way that
every rightmost vertex in is a vertex of ;
and is a vertex of .
Write for the set of global join plaquettes of the polygon .
There exists such that, for and any ,
For all and , there exist such that, for ,
The next lemma will be used in the proof of Proposition 4.5.
Let and . Writing so that , consider the two subpaths and , the first starting at and the second ending there. Each of these paths contains precisely one of the two horizontal edges of any element in .
Proof. For given , let . We may decompose as in accordance with Definition 4.4. We then have that is a vertex of and a vertex of . The path leaves to follow until passing through an edge in to arrive in , tracing this polygon until passing back through the other horizontal edge of and following until returning to . It is during the trajectory in that the visit to is made. ∎
Proof of Proposition 4.5. For , an element is called a half-space walk if for each . We call a half-space walk returning if, after the last visit that makes to the lowest -coordinate that this walk attains, makes a unique visit to the -axis, with this occurring at its endpoint . Let denote the set of length returning half-space walks. We will first argue that, for any ,
To see this, we consider a map . Let , and let be the index of the final visit of to the lowest -coordinate visited by . The image of is defined to be the concatenation of and the reflection of through the horizontal line that contains . (We have not defined concatenation but hope that the meaning is evident.) Our map is injective because, given an element in its image, the horizontal coordinate of the line used to construct the image walk may be read off from that walk; (the coordinate is one-half of the -coordinate of the image walk’s non-origin endpoint). Its image lies in the set of bridges of length , where a bridge is a walk whose starting point has maximal -coordinate and whose endpoint uniquely attains the minimal -coordinate. The set of bridges of length beginning at the origin has cardinality at most : this classical fact, which follows from (1.2.17) in  by symmetries of the Euclidean lattice, is proved by a superadditivity argument with similarities to the proof of (3.3). Thus, considering this map proves (4.3).
Noting these things allow us to reduce the proof of the proposition to verifying the following assertion. There exists such that, for , and any ,
Setting , we obtain Proposition 4.5.
To complete the proof of the proposition, we must prove (4.4), and this we now do. Let . Setting so that (as we did in Lemma 4.7), write and . Writing for reflection in the vertical (-directed) line that passes through , define a map to be the concatenation
By Lemma 4.7, each of and traverses precisely one horizontal edge of each of ’s global join plaquettes. Set and enumerate by the sequence (in an arbitrary order; for example, in the order in which traverses an edge of each plaquette). For each , let denote the unique edge in traversed by . Consider the path formed by modifying so that the one-step subpath is replaced by a three-step subpath from to that traverses the plaquette using its three edges other than . The modification may be made iteratively for several choices of , and the outcome is independent of the order in which the modifications are made. In this way, we may define a modified path for each , under which the modified route is taken along plaquettes precisely when . Note that is a self-avoiding walk whose length exceeds ’s by : it is self-avoiding because this walk differs from by several disjoint replacements of one-step subpaths by three-step alternatives, and, in each case, the two new vertices visited in the alternative route are vertices in , and, as such, cannot be vertices in .
Note further that the intersection of the edge-sets of and equals (where the sets in the union are each singletons).
For each , define ,
Recall that and that is given. Consider the multi-valued map
that associates to each with the set
where here we abuse notation and identify a subset of with its set of indices under the given enumeration of .
Note that, for some constant that is independent of and for all ,
(In the latter displayed inequality, the factor of that appears via Stirling’s formula has been cancelled against other omitted terms. This detail is inconsequential and the argument is omitted.)
Note that, for any , the preimage is either the empty-set or a singleton. Indeed, if for some with , then may be recovered from as follows:
the coordinate equals ;
the vertex is the lowest among the vertices of having the above -coordinate;
setting so that is this vertex, consider the non-self-avoiding walk . This walk begins and ends at . There are exactly instances where the walk traverses an edge twice. In each case, the three-step journey that the walk makes in the steps preceding, during and following the second crossing of the edge follow three edges of a plaquette. Replace this journey by the one-step journey across the remaining edge of the plaquette, in each instance. The result is .
4.3. Preparing for joining surgery: left and right polygons
When we prove Proposition 4.2 in the next subsection, polygons from a certain set will be joined to others in another set . We think of the former as being joined to the latter on the right, so that the superscripts indicate a handedness associated to the joining.
Anyway, in this subsection, we specify these polygon sets, give a lower bound on their size in Lemma 4.9, and then explain in Lemma 4.11 how there are plentiful opportunities for joining pairs of such polygons in a surgically useful way (so that the join may be detected by virtue of its global nature).
Let be a polygon. Recall from Definition 2.2 the notation and , as well as the height and width .
For , let denote the set of left polygons such that
(and thus, by a trivial argument, ),
Let denote the set of right polygons such that
Proof. An element not in is brought into this set by right-angled rotation. If, after the possible rotation, it is not in , it may brought there by reflection in the -axis. ∎
A Madras joinable polygon pair is called globally Madras joinable if the junction plaquette of the join polygon is a global join plaquette of .
Both polygon pairs in the upper part of Figure 2 are globally Madras joinable.
Let and let and .
is such that and some horizontal shift of is globally Madras joinable.
Write for the set of